An Invitation to Modern Number Theory

Steven J. Miller, Ramin Takloo-Bighash

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Description

In a manner accessible to beginning undergraduates, An Invitation to
Modern Number Theory introduces many of the central problems, conjectures,
results, and techniques of the field, such as the Riemann Hypothesis, Roth's
Theorem, the Circle Method, and Random Matrix Theory. Showing how experiments
are used to test conjectures and prove theorems, the book allows students to
do original work on such problems, often using little more than calculus
(though there are numerous remarks for those with deeper backgrounds). It
shows students what number theory theorems are used for and what led to them
and suggests problems for further research.

Steven Miller and Ramin Takloo-Bighash introduce the problems and the
computational skills required to numerically investigate them, providing
background material (from probability to statistics to Fourier analysis)
whenever necessary. They guide students through a variety of problems, ranging
from basic number theory, cryptography, and Goldbach's Problem, to the
algebraic structures of numbers and continued fractions, showing connections
between these subjects and encouraging students to study them further. In
addition, this is the first undergraduate book to explore Random Matrix
Theory, which has recently become a powerful tool for predicting answers in
number theory.

Providing exercises, references to the background literature, and Web links
to previous student research projects, An Invitation to Modern Number
Theory can be used to teach a research seminar or a lecture class.